Explanation

(c)
Given, $y = \cos^{2}{\frac{3x}{2} - \sin^{2}{\frac{3x}{2} = 2\cos^{2}{\frac{3x}{2} - 1}}}$
$\Rightarrow \ \ \ \frac{dy}{dx} = 2.2\cos{\frac{3x}{2}\left( - \sin\frac{3x}{2} \right)}\left( \frac{3}{2} \right)$
$\Rightarrow \ \ \ \frac{dy}{dx} = \ - 6\cos{\frac{3x}{2}\sin\frac{3x}{2}}$
$\Rightarrow \ \ \frac{d^{2}y}{dx^{2}} = \ - 6\ \left\lbrack \cos{\frac{3x}{2}\left( \cos\frac{3x}{2} \right)}.\frac{3}{2} - \sin{\frac{3x}{2}\sin{\frac{3x}{2}.\frac{3}{2}}} \right\rbrack$
$\Rightarrow \ \frac{d^{2}y}{dx^{2}} = \ - 9\left\lbrack \cos^{2}{\frac{3x}{2} - \sin^{2}\frac{3x}{2}} \right\rbrack = - 9y$
Alternate
$y = \cos^{2}{\frac{3x}{2} - \sin^{2}\frac{3x}{2}}$
$\Rightarrow \ y = \cos{3x}$
$\Rightarrow \ \frac{dy}{dx} = \ - 3\sin{3x}$
$\Rightarrow \ \frac{d^{2}y}{dx^{2}} = \ - 9\cos{3x = - 9y}$